Clique polynomials and independent set polynomials of graphs
Selected papers of the 13th British Combinatorial Conference on British combinatorial conference
On the hardness of approximate reasoning
Artificial Intelligence
Approximately counting up to four (extended abstract)
STOC '97 Proceedings of the twenty-ninth annual ACM symposium on Theory of computing
Randomized algorithms: approximation, generation, and counting
Randomized algorithms: approximation, generation, and counting
The complexity of counting colourings and independent sets in sparse graphs and hypergraphs
Computational Complexity
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
On Counting Independent Sets in Sparse Graphs
SIAM Journal on Computing
Corrigendum: the complexity of counting graph homomorphisms
Random Structures & Algorithms
Counting models for 2SAT and 3SAT formulae
Theoretical Computer Science
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The problem of counting the number of independent sets of a graph G (denoted as NI(G)) is a classic #P-complete problem for graphs of degree 3 or higher. Exploiting the strong relation between NI(G) and Fibonacci numbers, we show that if the depth-first graph of G does not contain a pair of basic cycles with common edges, then NI(G) can be computed in linear time (in the size of the graph). This determines new classes of instances of graphs without restrictions on their degrees and where the number of independent sets is computed in polynomial time. We design an exact deterministic algorithm for computing NI(G) based on the topological structure of the graph G, applying the well-known splitting rule from Davis and Putnam (D&P) procedure. D&P is a familiar method for solving the Satisfiability Boolean Problem. Our algorithm for computing NI(G) establishes a leading Worst-Case Upper Bound of O(poly(n,m)*1.220744n), n and m being the number of nodes and edges of the graph G, respectively. The exact technique reported here can be used to compute the redundancy of a line in a communication network.